Method and apparatus for selecting voltage and frequency levels for use in at-speed testing

ABSTRACT

In one embodiment, the invention is a method and apparatus for selecting voltage and frequency levels for use in at-speed testing. One embodiment of a method for selecting a set of test conditions with which to test an integrated circuit chip includes formulating a statistical optimization problem and obtaining a solution to the statistical optimization problem, where the solution is the set of test conditions.

BACKGROUND OF THE INVENTION

The present invention relates generally to design automation, andrelates more particularly to the at-speed structural test (ASST) ofintegrated circuit (IC) chips.

When IC chips come off the manufacturing line, the chips are tested“at-speed” to ensure that they perform correctly (and to filter outchips that do not perform correctly). In particular, the chips aretested at a specified voltage level (V_(test)). In addition, the chipsmust meet a specified frequency level (F_(test)) to be qualified as“good” chips.

The determination of V_(test) and F_(test) directly impacts the yield(i.e., the percentage of the chips that are shipped to customers) andthe shipped product quality loss (i.e., the percentage of the chips thatare shipped to customers that are “bad” chips). More stringent testconditions will improve shipped product quality loss but worsen yield.More relaxed test conditions will improve yield but worsen shippedproduct quality loss.

Conventionally, V_(test) and F_(test) are determined in a heuristicmanner based on empirical hardware characterization. This approach isperformed manually using engineering judgment. Thus, inherently, thisapproach has no mathematical formulation or statistical basis andrequires significant human effort. The resulting test conditions aretypically highly sensitive to variations in sample chips.

SUMMARY OF THE INVENTION

In one embodiment, the invention is a method and apparatus for selectingvoltage and frequency levels for use in at-speed testing. One embodimentof a method for selecting a set of test conditions with which to test anintegrated circuit chip includes formulating a statistical optimizationproblem and obtaining a solution to the statistical optimizationproblem, where the solution is the set of test conditions.

BRIEF DESCRIPTION OF THE DRAWINGS

So that the manner in which the above recited features of the presentinvention can be understood in detail, a more particular description ofthe invention may be had by reference to embodiments, some of which areillustrated in the appended drawings. It is to be noted, however, thatthe appended drawings illustrate only typical embodiments of thisinvention and are therefore not to be considered limiting of its scope,for the invention may admit to other equally effective embodiments.

FIG. 1 is a flow diagram illustrating one embodiment of a method forselecting voltage and frequency levels for use in at-speed testing of aset of integrated circuit chips, according to the present invention;

FIG. 2 is a graph illustrating voltage versus highest working frequencyfor a set of four sample chips; and

FIG. 3 is a high-level block diagram of test condition selection methodthat is implemented using a general purpose computing device.

DETAILED DESCRIPTION

In one embodiment, the present invention is a method and apparatus forselecting voltage and frequency levels for use in at-speed testing of ICchips. Embodiments of the invention provide an automated technique fordetermining the optimal voltage and frequency levels for testing a setof IC chips. The voltage and frequency levels are then applied by atester (i.e., a system or device that electronically tests the IC chipsunder a set of test parameters in order to determine whether theiroperational characteristics match simulation models or otherdocumentation for the IC chips). The present invention achieves betterand more robust testing of IC chips, as well as higher productivity,than is possible with conventional heuristic practices.

FIG. 1 is a flow diagram illustrating one embodiment of a method 100 forselecting voltage and frequency levels for use in at-speed testing of aset of IC chips, according to the present invention. The method 100 maybe implemented, for example, at a processor that is coupled to a testerfor testing the set of IC chips. Alternatively, the method 100 may beimplemented at the tester.

The method 100 is initialized at step 102 and proceeds to step 104,where the method 100 obtains a set of k sample chips from the set of NIC chips (where k≦N). The k sample chips can be selected randomly orselected over a set of parametric indicators (for example, over a rangeof performance-sensitive ring oscillator (PSRO) frequencies over the setof N IC chips).

In step 106, the method 100 shmoos (i.e., plots the response varyingover a range of conditions and inputs) the set of sample chips. In oneembodiment, shmooing involves, for each of the k sample chips, applyinga set of voltages over some range of voltages. At each applied voltage,the frequency of the sample chip is varied until the sample chip fails.Thus, the highest working frequency (i.e., the highest operatingfrequency at which the sample chip does not fail) is obtained for eachsample chip under different voltage levels.

FIG. 2, for example, is a graph illustrating voltage versus the highestworking frequency, with the domain frequency (i.e., the frequency usedby customers in normal operation) illustrated as a horizontal line for aset of four sample chips (designated as SAMPLE CHIP 1, SAMPLE CHIP 2,SAMPLE CHIP 3, and SAMPLE CHIP 4). The PSRO frequency for each samplechip is indicated in parentheses; as illustrated, the sample chips covera range of PSRO frequencies from 151 nanoseconds to 219 nanoseconds.Each point on the graph designates, for a given sample chip, the highestworking frequency at a corresponding voltage.

Referring back to FIG. 1, in step 108, the method 100 runs a regressionon the highest working frequencies obtained in step 106. The result ofthe regression is a mathematical model that captures the relationbetween the working frequencies, the characteristics of the set ofsample chips, and the different voltage levels.

In step 110, the method 100 uses the regression data to solve astatistical optimization problem. The solution to the optimizationproblem comprises the optimal testing conditions for the frequency andthe voltage (i.e., F_(test) and V_(test)). In particular, theoptimization problem attempts to achieve a target scrap rate, T_(scrap)(i.e., a percentage of the set of N IC chips that is scrapped duringASST) based on the mathematical model obtained in step 108. The scraprate is typically used by test engineers as an indicator to judge thetesting quality. When the scrap rate is too high, too many chips arediscarded, and yield suffers. When the scrap rate is too low, too manychips are shipped to customers, and shipped product quality loss may beat risk of being high. In one embodiment, T_(scrap) is in the range of0.5 to 2.5 percent.

In one embodiment, the optimization problem seeks to select values forF_(test) and V_(test) that minimize the deviation from the target scraprate. Thus, in one embodiment, the optimization problem is formulated asfollows:

$\begin{matrix}\min \\{F_{test},V_{test}}\end{matrix}{{{P( {{F( {{PSRO},V_{test}} )} \leq F_{test}} )} - T_{scrap}}}$

such that:

V_(min)≦V_(test)≦V_(max)

F_(min)≦F_(test)≦F_(max)

F(PSRO_(k),V),kε[1,N]ε[V_(min),V_(max)]  (EQN. 1)

where PSRO is the PSRO frequency for all chips in the set of N IC chips,F(PSRO, V) is the modeled highest working frequency for all chips withdifferent PSRO frequencies at any voltage level V, F(PSRO_(k), V) arethe shmooed highest working frequencies for the set of k sample chips(e.g., in the case of FIG. 2, the k=4 slanted curves),P(F(PSRO,V_(test))≦F_(test)) is the probability that the frequency forall chips in the set of N IC chips as a function of PSRO frequency andtest voltage V_(test) is less than or equal to the test frequencyF_(test), V_(min) and V_(max) are the minimum and maximum allowablevoltage levels, respectively, and F_(min) and F_(max) are the minimumand maximum achievable frequencies in the tester, respectively.

The modeled F(PSRO, V) in the statistical optimization problem definedin EQN. 1 comprises a known (analytical) part and an unknown part. Theknown part has a given functional form, while the unknown part ismodeled as one or more random variables. In one embodiment, the unknownpart of the optimization problem is modeled as a random variable thatrepresents variation. For example, F(PSRO, V) may be modeled as follows:

F(PSRO,V)=G(PSRO,V)+ΔR  (EQN. 2)

where R is the random variable.

The known part of F(PSRO, V) can be any user-selected functional formbased on engineering knowledge about the process. For example, the knownpart of the optimization problem can be modeled as:

G(PSRO,V)=α₀+(α₁·PSRO)+(α₂ ·V)  (EQN. 3)

where the frequency F is a linear function of the voltage V and the PSROfrequency.

The regression performed in step 108 provides the known part of F(PSRO,V), based on the shmoo data. The remaining error or uncertainty isassigned to the unknown random variable.

In one embodiment, the PSRO frequency can be modeled as a distribution.In another embodiment, the random variation ΔR can be adjusted to makethe model more comfortably capture model uncertainty. This adjustmentcan be based on the sample chip data and/or on expert or domainknowledge.

For example, referring back to EQN. 1 by assuming Gaussian randomdistributions for both PSRO and ΔR, EQN. 1 could be alternativelyformulated as:

$\begin{matrix}{\begin{matrix}\min \\{F_{test},V_{test}}\end{matrix}{{{\Phi ( {{PSRO}, {\Delta \; R} \middle| V_{test} ,F_{test}} )} - T_{scrap}}}} & ( {{EQN}.\mspace{14mu} 4} )\end{matrix}$

More specifically, solving EQN. 1 based on a linear regression model,one can obtain:

$\begin{matrix}{\mspace{79mu} \begin{matrix}{{F( {{PSRO},V} )} = {{G( {{PSRO},V} )} + {\Delta \; R}}} \\{= {\alpha_{0} + ( {\alpha_{1} \cdot {PSRO}} ) + ( {\alpha_{2} \cdot V} ) + {\Delta \; R}}}\end{matrix}} & ( {{EQN}.\mspace{14mu} 5} ) \\{\mspace{79mu} {{Thus},}} & \; \\\begin{matrix}{{P( {{F( {{PSRO},V_{test}} )} \leq F_{test}} )} = {\Phi ( \frac{F_{test} - \mu_{F}}{\sigma_{F}} )}} \\{= {\Phi ( \frac{\begin{matrix}{F_{test} - \alpha_{0} - ( {\alpha_{1} \cdot \mu_{{PSRO}\;}} ) +} \\{( {\alpha_{2} \cdot V_{test}} ) - \mu_{\Delta \; R}}\end{matrix}}{\sqrt{( {\alpha_{1}^{2} \cdot \sigma_{PSRO}^{2}} )} + \sigma_{\Delta \; R}^{2}} )}} \\{= T_{{scrap}\;}}\end{matrix} & ( {{EQN}.\mspace{14mu} 6} ) \\{\mspace{79mu} {{And},}} & \; \\{{\Phi^{- 1}( T_{scrap} )} = \frac{F_{test} - \alpha_{0} - ( {\alpha_{1} - \mu_{PSRO}} ) + ( {\alpha_{2} \cdot V_{test}} ) - \mu_{\Delta \; R}}{\sqrt{( {\alpha_{1}^{2} \cdot \sigma_{PSRO}^{2}} )} + \sigma_{\Delta \; R}^{2}}} & ( {{EQN}.\mspace{14mu} 7} )\end{matrix}$

For the same target scrap rate, one can have multiple choices ofV_(test) and F_(test) within a valid region. These choices fall within astraight line bounded by [V_(min), V_(max)] and [F_(min), F_(max)]. Inone embodiment, F_(min) is chosen as F_(domain). Thus, F_(test) can besolved as:

$\begin{matrix}{F_{test} = {{\alpha_{2} \cdot V_{test}} + \alpha_{0} + ( {\alpha_{1} \cdot \mu_{PSRO}} ) + \mu_{\Delta \; R} + {{\Phi^{- 1}( T_{scrap} )} \cdot \sqrt{( {{\alpha_{1}^{2} \cdot \sigma_{PSRO}^{2}} + \sigma_{\Delta \; R}^{2}} )}}}} & ( {{EQN}.\mspace{14mu} 8} )\end{matrix}$

The method 100 outputs the optimal testing conditions F_(test) andV_(test) in step 112. In one embodiment, the optimal testing conditionsare output directly to the tester (e.g., by programming the tester),which applies the optimal testing conditions to testing of the set of ICchips. The method 100 then terminates in step 114.

The method 100 therefore provides a framework to systematically examinedifferent effects on chosen testing conditions. Application of themethod 100 produces high-quality testing results that are robust tosample variations. Moreover, the automated flow of the present inventionenhanced the productivity of the testing.

In one embodiment, a more general formulation of EQN. 8 can be used toextend the present invention to IC chips having multiple clock domains.When there are multiple clock domains, each clock domain would have itsown model:

∀iε[1,m]:F _(i)(PSRO,V)=α_(i,0)+(α_(i,1)·PSRO)+(α_(i,2) ·V)+ΔR_(i)  (EQN. 9)

The final scrap rate should be a joint scrap rate for all clock domains,i.e.:

P(F ₁(PSRO,V _(1,test))≦F _(1,test), or . . . , or F _(m)(PSRO,V_(m,test))≦F _(m,test))  (EQN. 10)

The optimization problem can thus be similarly formulated as:

$\begin{matrix}\min \\{F_{i,{test}},V_{i,{test}}}\end{matrix}{{\begin{matrix}{P( {{{F_{1}( {{PSRO},V_{1,{test}}} )} \leq F_{1,{test}}},{{or}\mspace{14mu} \ldots}\mspace{14mu},} } \\{ {{{or}\mspace{14mu} {F_{m}( {{PSRO},V_{m,{test}}} )}} \leq F_{m,{test}}} ) - T_{scrap}}\end{matrix}}.}$

such that

∀iε[1,m]

V_(min)≦V_(i,test)≦V_(max)

F_(i,min)≦F_(i,test)≦F_(max)  (EQN. 11)

EQN. 11 can be solved numerically, such that the optimal solution is notunique and may not necessarily form a hyper plane. Alternatively, EQN.11 can be solved iteratively. In the latter case, the test conditionsfor each clock domain are independently found to target an adjustedscrap rate that is lower than the target scrap rate. The same proceduresdescribed above with respect to FIG. 1 are followed for each clockdomain. The joint scrap rate is then computed to determine what jointscrap rate can be finally achieved (e.g., using EQN. 10). If the jointscrap rate is higher than the target scrap rate, the adjusted scrap rateis reduced; if the joint scrap rate is lower than the target scrap rate,the adjusted scrap rate is increased.

The present invention therefore makes use of all shmoo data generatedfrom a set of sample chips. The end result is thus less sensitive to oneparticular sample chip. In addition, by statistically targeting the“desired” scrap rate directly, the chances of over-scrapping orunder-scrapping can be minimized.

FIG. 3 is a high-level block diagram of test condition selection methodthat is implemented using a general purpose computing device 300. In oneembodiment, a general purpose computing device 300 comprises a processor302, a memory 304, a test condition selection module 305 and variousinput/output (I/O) devices 306 such as a display, a keyboard, a mouse, astylus, a wireless network access card, and the like. In one embodiment,at least one I/O device is a storage device (e.g., a disk drive, anoptical disk drive, a floppy disk drive, a path selection tool, and/or atest pattern generation tool). It should be understood that the testcondition selection module 305 can be implemented as a physical deviceor subsystem that is coupled to a processor through a communicationchannel.

Alternatively, the test condition selection module 305 can berepresented by one or more software applications (or even a combinationof software and hardware, e.g., using Application Specific IntegratedCircuits (ASIC)), where the software is loaded from a storage medium(e.g., I/O devices 306) and operated by the processor 302 in the memory304 of the general purpose computing device 300. Thus, in oneembodiment, the test condition selection module 305 for selecting avoltage and frequency levels for use in at-speed structural testing ofan IC chip, as described herein with reference to the preceding Figures,can be stored on a non-transitory computer readable storage medium(e.g., RAM, magnetic or optical drive or diskette, and the like).

It should be noted that although not explicitly specified, one or moresteps of the methods described herein may include a storing, displayingand/or outputting step as required for a particular application. Inother words, any data, records, fields, and/or intermediate resultsdiscussed in the methods can be stored, displayed, and/or outputted toanother device as required for a particular application. Furthermore,steps or blocks in the accompanying Figures that recite a determiningoperation or involve a decision, do not necessarily require that bothbranches of the determining operation be practiced. In other words, oneof the branches of the determining operation can be deemed as anoptional step.

While the foregoing is directed to embodiments of the present invention,other and further embodiments of the invention may be devised withoutdeparting from the basic scope thereof. Various embodiments presentedherein, or portions thereof, may be combined to create furtherembodiments. Furthermore, terms such as top, side, bottom, front, back,and the like are relative or positional terms and are used with respectto the exemplary embodiments illustrated in the figures, and as suchthese terms may be interchangeable.

1. A method for selecting a set of test conditions with which to test anintegrated circuit chip, the method comprising: formulating astatistical optimization problem; and obtaining a solution to thestatistical optimization problem, the solution comprising the set oftest conditions, wherein at least one of the formulating and theobtaining is performed by a processor.
 2. The method of claim 1, whereinthe set of test conditions comprises: a test voltage and a testfrequency.
 3. The method of claim 1, wherein the statisticaloptimization problem seeks to minimize deviation from a target scraprate for a set of chips including the integrated circuit chip.
 4. Themethod of claim 1, wherein a highest working frequency model for allchips in a set of chips including the integrated circuit chip comprises:a known part; and an unknown part.
 5. The method of claim 4, wherein theunknown part is modeled as one or more random variables.
 6. The methodof claim 4, wherein the known part is solved using a mathematical modelthat describes a set of sample chips, where the set of sample chips isselected from a set of chips including the integrated circuit chip. 7.The method of claim 6, wherein the mathematical model describes, foreach chip in the set of sample chips, a relationship between highestworking frequency, voltage, and chip characteristics.
 8. The method ofclaim 6, wherein the mathematical model is obtained by: shmooing the setof sample chips; and running a regression on results of the shmooing. 9.The method of claim 8, wherein the shmooing comprises: defining a set ofvoltages; obtaining a highest working frequency at each voltage in theset of voltages, for each sample chip in the set of sample chips. 10.The method of claim 1, further comprising: outputting, by the processor,the set of test conditions to a tester; and programming the tester toapply the set of test conditions during testing of the integratedcircuit chip.
 11. The method of claim 1, wherein the processor is partof a tester that applies the set of test conditions during testing ofthe integrated circuit chip.
 12. A non-transitory computer readablemedium containing an executable program for selecting a set of testconditions with which to test an integrated circuit chip, where theprogram performs the steps of: formulating a statistical optimizationproblem; and obtaining a solution to the statistical optimizationproblem, the solution comprising the set of test conditions.
 13. Thenon-transitory computer readable medium of claim 12, wherein the set oftest conditions comprises: a test voltage and a test frequency.
 14. Thenon-transitory computer readable medium of claim 12, wherein thestatistical optimization problem seeks to minimize deviation from atarget scrap rate for a set of chips including the integrated circuitchip.
 15. The non-transitory computer readable medium of claim 12,wherein a highest working frequency model for all chips in a set ofchips including the integrated circuit chip comprises: a known part; andan unknown part.
 16. The non-transitory computer readable medium ofclaim 15, wherein the unknown part is modeled as one or more randomvariables.
 17. The non-transitory computer readable medium of claim 15,wherein the known part is solved using a mathematical model thatdescribes a set of sample chips, where the set of sample chips isselected from a set of chips including the integrated circuit chip. 18.The non-transitory computer readable medium of claim 17, wherein themathematical model is obtained by: shmooing the set of sample chips; andrunning a regression on results of the shmooing.
 19. The non-transitorycomputer readable medium of claim 18, wherein the shmooing comprises:defining a set of voltages; obtaining a highest working frequency ateach voltage in the set of voltages, for each sample chip in the set ofsample chips.
 20. Apparatus for selecting a set of test conditions withwhich to test an integrated circuit chip, the apparatus comprising:means for formulating a statistical optimization problem; and means forobtaining a solution to the statistical optimization problem, thesolution comprising the set of test conditions.